$\mathcal{L}$-invariants and exceptional zeros of Bianchi modular forms

TL;DR

This paper proves an exceptional zero conjecture for Bianchi modular forms over imaginary quadratic fields, establishing the existence of L-invariants and relating derivatives of p-adic L-functions to classical L-values, extending known results to new settings.

Contribution

It introduces L-invariants for Bianchi modular forms and proves their role in exceptional zero phenomena, generalizing Darmon and Orton's methods to imaginary quadratic fields.

Findings

Existence of L-invariants for Bianchi modular forms at primes above p.

Relation between derivatives of p-adic L-functions and classical L-values involving L-invariants.

Connection of L-invariants to classical modular forms when p is inert, confirming a conjecture.

Abstract

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime $\mathfrak{p}$ of F above p we prove the existence of an L-invariant $\mathcal{L}_{\mathfrak{p}}$, depending only on $\mathfrak{p}$ and f, such that when the p-adic L-function of f has an exceptional zero at $\mathfrak{p}$, its derivative can be related to the classical L-value multiplied by $\mathcal{L}_{\mathfrak{p}}$. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL(2) over the rationals. When p is not split and f is the base-change of a classical modular form F, we relate $\mathcal{L}_{\mathfrak{p}}$ to the L-invariant of F, resolving a conjecture of Trifkovi\'{c} in this case.